We say that a lattice Λ \Lambda in n n -dimensional Euclidean space E n {E_n} provides a k k -fold packing for spheres of radius 1 if, when open spheres of radius 1 are centered at the points of Λ \Lambda , no point of space lies in more than k k spheres. The multiple packing constant Δ k ( n ) \Delta _k^{(n)} is the smallest determinant of any lattice with this property. In the plane, the first three multiple packing constants Δ 2 ( 2 ) , Δ 3 ( 2 ) \Delta _2^{(2)},\Delta _3^{(2)} , and Δ 4 ( 2 ) \Delta _4^{(2)} are known, due to the work of Blundon, Few, and Heppes. In E 3 , Δ 2 ( 3 ) {E_3},\Delta _2^{(3)} is known, because of work by Few and Kanagasabapathy, but no other multiple packing constants are known. We show that Δ 3 ( 3 ) ⩽ 8 38 / 27 \Delta _3^{(3)} \leqslant 8\sqrt {38} /27 and give evidence that Δ 3 ( 3 ) = 8 38 / 27 \Delta _3^{(3)} = 8\sqrt {38} /27 . We show, in fact, that a lattice with determinant 8 38 / 27 8\sqrt {38} /27 gives a local minimum of the determinant among lattices providing a 3 3 -fold packing for the unit sphere in E 3 {E_3} .